Bayes error rate formula clarification
Bayes Error Rate
For the general case of K different classes, the probability of classifing x instance correctly is:
\begin{equation} \label{eq1}
\begin{split}
P(correct) &amp; = \sum_{i=1}^{K} p(x \in H_i, C_i) \\
 &amp; = \sum_{i=1}^{K} \int_{x \in H_i} p(x,C_i) \, dx\\
 &amp; = \sum_{i=1}^{K} \int_{x \in H_i} P(C_i|x)p(x)\,dx \\
\end{split}
\end{equation}
where $H_i$ is the region where class $i$ has the highest posterior. So the Bayes Error Rate is:
\begin{equation} \label{eq2}
\begin{split}
P(error) &amp; = 1 - p(correct) \\
 &amp; = 1 - \sum_{i=1}^{K} \int_{x \in H_i} P(C_i|x)p(x)\,dx
\end{split}
\end{equation}
Be careful if we drew $x$ from $H_i$, the region covered/classified by $C_i$, $P(C_i|x) \ne 1$, because that would mean that we surely predict correctly all the time. If $x$ belongs to a decision area, this does not imply that it belongs to the corresponding class.